From Zero to Engineer

Arithmetic - review exercises

A set of exercises from the first part of the course: inequality signs, order of operations, prime factors, rounding to tens, hundreds and thousands, GCD and LCM, and operations on common fractions.

This is a review of all the arithmetic the course has covered so far. The exercises are short, but each one touches a different skill: comparing negative numbers, the order of operations, prime factorisation, rounding, GCD and LCM, and calculations with fractions. If one of them gives you trouble, go back to the lesson it comes from - the numbers are in the course contents.

Arithmetic - review exercises I

Insert the appropriate sign, < or >, between each of the following pairs of numbers.

\text{a)}\quad -12 > -15

\text{b)}\quad 9 > -17

\text{c)}\quad -11 < 10

Evaluate each of the following expressions.

\begin{aligned}\text{a)}\quad 24 - 3 \cdot 4 + 28 : 14 &= 24 - 12 + 2 \\&= 12 + 2 = 14\end{aligned}

\begin{aligned}\text{b)}\quad (24 - 3) \cdot (4 + 28) : 14 &= 21 \cdot 32 : 14 \\&= 672 : 14 = 48\end{aligned}

Express each of the following numbers as a product of prime factors.

156	2
78	2
39	3
13	13
1

156 = 2 \cdot 2 \cdot 3 \cdot 13

546	2
273	3
91	7
13	13
1

546 = 2 \cdot 3 \cdot 7 \cdot 13

1445	5
289	17
17	17
1

1445 = 5 \cdot 17 \cdot 17

1485	3
495	3
165	3
55	5
11	11
1

1485 = 3 \cdot 3 \cdot 3 \cdot 5 \cdot 11

Round each of the following numbers to the nearest 10, 100 and 1,000.

number	to 10	to 100	to 1,000
5045	5050	5000	5000
1100	1100	1100	1000
−1552	−1550	−1600	−2000
−4995	−5000	−5000	−5000

Find the GCD and LCM of the following numbers.

a) 1274 and 195

1274	2
637	7
91	7
13	13
1

1274 = 2 \cdot 7 \cdot 7 \cdot 13

195	3
65	5
13	13
1

195 = 3 \cdot 5 \cdot 13

\mathrm{GCD}(1274;\ 195) = 13 = 13

\mathrm{LCM}(1274;\ 195) = 2 \cdot 7 \cdot 7 \cdot 13 \cdot 3 \cdot 5 = 19110

b) 64 and 18

64	2
32	2
16	2
8	2
4	2
2	2
1

64 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

18	2
9	3
3	3
1

18 = 2 \cdot 3 \cdot 3

\mathrm{GCD}(64;\ 18) = 2 = 2

\mathrm{LCM}(64;\ 18) = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 576

Reduce each of the following fractions to its simplest form.

\text{a)}\quad \frac{8}{14} = \frac{8 : 2}{14 : 2} = \frac{4}{7}

\text{b)}\quad \frac{162}{36} = \frac{162 : 18}{36 : 18} = \frac{9}{2}

\text{c)}\quad -\frac{279}{27} = -\frac{279 : 9}{27 : 9} = -\frac{31}{3}

\text{d)}\quad -\frac{81}{3} = -\frac{81 : 3}{3 : 3} = -\frac{27}{1} = -27

Evaluate each of the following expressions, giving your answer as a common fraction.

\text{a)}\quad \frac{1}{3} + \frac{3}{5} = \frac{1}{3} \cdot \frac{5}{5} + \frac{3}{5} \cdot \frac{3}{3} = \frac{5}{15} + \frac{9}{15} = \frac{14}{15}

\text{b)}\quad \frac{2}{7} - \frac{1}{9} = \frac{2}{7} \cdot \frac{9}{9} - \frac{1}{9} \cdot \frac{7}{7} = \frac{18}{63} - \frac{7}{63} = \frac{11}{63}

\text{c)}\quad \frac{8}{3} \cdot \frac{6}{5} = \frac{48}{15} = \frac{48 : 3}{15 : 3} = \frac{16}{5}

\displaystyle \frac{4}{5} \cdot \frac{2}{15} = \frac{8}{75}

d) four fifths of two fifteenths

\text{e)}\quad \frac{9}{2} : \frac{3}{2} = \frac{9}{2} \cdot \frac{2}{3} = \frac{3}{1} = 3

\begin{aligned}\text{f)}\quad \frac{6}{7} - \frac{4}{5} \cdot \frac{3}{2} : \frac{7}{5} + \frac{9}{4} &= \frac{6}{7} - \frac{12}{10} \cdot \frac{5}{7} + \frac{9}{4} \\&= \frac{6}{7} - \frac{12}{14} + \frac{9}{4} \\&= \frac{6}{7} \cdot \frac{4}{4} - \frac{12}{14} \cdot \frac{2}{2} + \frac{9}{4} \\&= \frac{24}{28} - \frac{24}{28} + \frac{9}{4} = \frac{9}{4}\end{aligned}

Frequently asked questions

Which negative number is the larger one?

The one closer to zero. That is why −12 > −15, even though 15 is more than 12 - with negative numbers a larger absolute value means a smaller number.

In what order are operations carried out?

Brackets first, then multiplication and division, and addition and subtraction last. That is why 24 − 3 · 4 + 28 : 14 is 24 − 12 + 2, which is 14.

How do you read the GCD and LCM from the prime factors?

The GCD is the product of the factors common to both numbers, and the LCM is the product of all the factors of one number plus those of the other that have not been used yet. For 1274 and 195 only thirteen is common, so the GCD is 13 and the LCM is 19,110.

What is 5045 rounded to the nearest ten?

5050. The final five is exactly halfway between 5040 and 5050, and a half is rounded up, that is away from zero. The same number to the nearest hundred gives 5000.

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Number systems - review exercises

Arithmetic - review exercises II